As shown in Figure 2.3 the earth axis of rotation (the polar axis) is always inclined at an angle of 23.45° from the ecliptic axis, which is normal to the ecliptic plane. The ecliptic plane is the plane of orbit of the earth around the sun. As the earth rotates around the sun it is as if the polar axis is moving with respect to the sun. The solar declination is the angular distance of the sun's rays north (or south) of the equator, north declination designated as positive. As shown in Figure 2.5 it is the angle between the sun-earth center line and the projection of this line on the equatorial plane. Declinations north of the equator (summer in the Northern Hemisphere) are positive, and those south are negative. Figure 2.6 shows the declination during the equinoxes and the solstices. As can be seen, the declination ranges from 0° at the spring equinox to + 23.45° at the summer solstice, 0° at the fall equinox, and -23.45° at the winter solstice.
FIGURE 2.6 Yearly variation of solar declination.
FIGURE 2.6 Yearly variation of solar declination.
Jan Feb March April May June July Aug Sept Oct Nov Dec
Jan Feb March April May June July Aug Sept Oct Nov Dec
Day number
FIGURE 2.7 Declination of the sun.
Day number
The variation of the solar declination throughout the year is shown in Figure 2.7. The declination, 6, in degrees for any day of the year (N) can be calculated approximately by the equation (ASHRAE, 2007)
Declination can also be given in radians1 by the Spencer formula (Spencer, 1971):
8 = 0.006918 - 0.399912cos(r) + 0.070257sin(r) 0.006758 cos(2r) + 0.000907 sin(2r) 0.002697 cos(3r) + 0.00148 sin(3r) (2.6)
where r is called the day angle, given (in radians) by r = 2n( N -1) (2.7)
The solar declination during any given day can be considered constant in engineering calculations (Kreith and Kreider, 1978; Duffie and Beckman, 1991).
1 Radians can be converted to degrees by multiplying by 180 and dividing by tc.
Month |
Day number |
Average day of the month | ||
Date |
N |
6 (deg.) | ||
January |
i |
17 |
17 |
-20.92 |
February |
31 + i |
16 |
47 |
-12.95 |
March |
59 + i |
16 |
75 |
-2.42 |
April |
90 + i |
15 |
105 |
9.41 |
May |
120 + i |
15 |
135 |
18.79 |
June |
151 + i |
11 |
162 |
23.09 |
July |
181 + i |
17 |
198 |
21.18 |
August |
212 + i |
16 |
228 |
13.45 |
September |
243 + i |
15 |
258 |
2.22 |
October |
273 + i |
15 |
288 |
-9.60 |
November |
304 + i |
14 |
318 |
-18.91 |
December |
334 + i |
10 |
344 |
-23.05 |
As shown in Figure 2.6, the Tropics of Cancer (23.45°N) and Capricorn (23.45°S) are the latitudes where the sun is overhead during summer and winter solstice, respectively. Another two latitudes of interest are the Arctic (66.5°N) and Antarctic (66.5°S) Circles. As shown in Figure 2.6, at winter solstice all points north of the Arctic Circle are in complete darkness, whereas all points south of the Antarctic Circle receive continuous sunlight. The opposite is true for the summer solstice. During spring and fall equinoxes, the North and South Poles are equidistant from the sun and daytime is equal to nighttime, both of which equal 12 h.
Because the day number and the hour of the year are frequently required in solar geometry calculations, Table 2.1 is given for easy reference.
HOUR ANGLE, h
The hour angle, h, of a point on the earth's surface is defined as the angle through which the earth would turn to bring the meridian of the point directly under the sun. Figure 2.5 shows the hour angle of point P as the angle measured on the earth's equatorial plane between the projection of OP and the projection of the sun-earth center to center line. The hour angle at local solar noon is zero, with each 360/24 or 15° of longitude equivalent to 1 h, afternoon hours being designated as positive. Expressed symbolically, the hour angle in degrees is h = ±0.25 (Number of minutes from local solar noon) (2.8)
where the plus sign applies to afternoon hours and the minus sign to morning hours.
The hour angle can also be obtained from the apparent solar time (AST); i.e., the corrected local solar time is h = (AST - 12)15 (2.9)
At local solar noon, AST = 12 and h = 0°. Therefore, from Eq. (2.3), the local standard time (the time shown by our clocks at local solar noon) is
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